On the reducibility of a class of nonlinear quasi-periodic system with small perturbation parameter near zero equilibrium point

Abstract

This work focuses on the reducibility of the following real nonlinear analytical quasiperiodic system: x ̇ = A x + f ( t , x , ϵ ) , x ∈ R 2 where A is a real 2×2 constant matrix, and f ( t , 0 , ϵ ) = O ( ϵ ) and ∂ x f ( t , 0 , ϵ ) = O ( ϵ ) as ϵ → 0 . With some non-resonant conditions of the frequencies with the eigenvalues of A and without any nondegeneracy condition with respect to ϵ , by an affine analytic quasiperiodic transformation we change the system to a suitable norm form at the zero equilibrium for most of the sufficiently small perturbation parameter ϵ .